A farmer collected data on the annual rainfall, x cm, and the annual yield of peas, p tonnes per acre. The data for annual rainfall was coded using $v = \frac{x - 5}{10}$ and the following statistics were found. $S_w = 5.753$ $S_p = 1.688$ $S_{pw} = 1.168$ $\bar{p} = 3.22$ $\bar{v} = 4.42$ (a) Find the equation of the regression line of p on v in the form $p = a + bv$. (b) Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm.

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A farmer collected data on the annual rainfall, x cm, and the annual yield of peas, p tonnes per acre - Edexcel - A-Level Maths Statistics - Question 4 - 2011 - Paper 1

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A farmer collected data on the annual rainfall, x cm, and the annual yield of peas, p tonnes per acre. The data for annual rainfall was coded using $v = \frac{x - 5... show full transcript

Worked Solution & Example Answer:A farmer collected data on the annual rainfall, x cm, and the annual yield of peas, p tonnes per acre - Edexcel - A-Level Maths Statistics - Question 4 - 2011 - Paper 1

Step 1

Find the equation of the regression line of p on v in the form $p = a + bv$

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Answer

To find the regression line, we first need to determine the slope (b) and the y-intercept (a).

The slope b is calculated using the formula: $b = \frac{S_{pw}}{S_v}$

Substituting the values: $b = \frac{1.168}{5.753} \approx 0.203$

Next, we calculate the intercept a using: $a = \bar{p} - b \bar{v}$

Substituting in the known values: $a = 3.22 - 0.203 \cdot 4.42 \approx 3.22 - 0.896 \approx 2.324$

Thus, the equation of the regression line is: $p = 2.324 + 0.203v$

Step 2

Using your regression line estimate the annual yield of peas per acre when the annual rainfall is 85 cm

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Answer

To estimate the yield when $x = 85$ cm, we first compute the coded value v:

$v = \frac{85 - 5}{10} = \frac{80}{10} = 8$

Next, substitute v into the regression equation: $p = 2.324 + 0.203 \cdot 8$

Calculating this gives: $p = 2.324 + 1.624 \approx 3.948$

Therefore, the estimated annual yield of peas per acre when the annual rainfall is 85 cm is approximately 3.95 tonnes.

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